Beware sequels. Neil MacGregor's radio series and accompanying book, A History of the World in 100 Objects, inspired by his stewardship of the British Museum, was acclaimed for its imaginative daring. What to do for an encore? You can almost hear the cogs of the commissioning process grinding: world history  world literature  global audience  global literary brand ? Shakespeare. Bingo!

MacGregor had already proved himself a brilliant "snapper-up of unconsidered trifles", Shakespeare's description of light-fingered Autolycus in The Winter's Tale. Why not let him choose 20 secular relics from around 1600, attach the Bard's name to the project, sit back and watch the ratings come home to roost?

This beautifully produced souvenir volume, a collaboration between the BM, Radio 4 and the publishers, works nicely enough as popular history for middle England. MacGregor, who confesses great reverence for what he calls "the charisma of things", has made some provocative picks from the national collection: a tiny model of a Scottish ship, said to propitiate witches; John Dee's magic mirror; the fighting kit of Henry V; a "German" clock; the preserved "eye" of the martyred Edward Oldcorne; the "Robben Island Bible" (a collected Shakespeare) inscribed by Nelson Mandela during his long incarceration.

This last is a bit of a giveaway. Mandela is no Tudor or Jacobean superhero. One flaw of this book  presumably not a problem in the more immediate medium of radio  is that MacGregor wants to have it both ways. As a museum mole he is eager to drill down into the forgotten, Elizabethan detail of his chosen artefacts. At the same time he wants to suggest that Shakespeare's "restless world" is a mirror to our own.

This pardonable delusion is often held by those who believe that not much is known about the author of Hamlet, The Tempest and the Sonnets. A sentimental attachment to the "mystery" of Shakespeare goes hand in hand with the idea that he and his works are a sublime mirror to our contemporary predicament. But the truth is that we know a surprising amount about Shakespeare, especially as a man of property and an enthusiastic litigant. Moreover, those universal themes are a sleight of hand. His late Tudor, early Jacobean world is strikingly different from our own  in faith, in science, in culture, politics and economics.

Never mind. MacGregor describes Shakespeare's England as "going global" with Francis Drake. London becomes the setting for Sex and the City. (What city does not have sex?) Ireland is the scene for "failing in the present". There's even a topical reference to the intractable politics of the union, "the flag that failed", in which MacGregor is forced to concede that the playwright "on the whole evades the question of Britain". (It's not obvious he ever cared enough about this question to "evade" it.)

You don't have to have an arts diploma to see that this is history as advertising. Francis Drake was never an astronaut. It does not help to compare Puck putting "a girdle round the earth in forty minutes" to a modern satellite. Venice's red-light district may have been "another dimension of Venetian freedom", but it was never Manhattan. MacGregor's claim that Othello and The Merchant of Venice "would not have worked, could not have been imagined, if they had been set in London" is a line that plays better in the studio than on the page.

These are blemishes, not flaws. When MacGregor the museum curator sticks to what he knows  museum objects  he can be a revelation. Keyed into gobbets of text, his choice of clocks, mirrors and swords etc opens a door on to the lost world of London's theatregoers in and around 1600.

MacGregor calls it "restless", but "terrifying" might be more accurate. Shakespeare and his contemporaries lived in the shadow of the scaffold, the gallows and the plague. The fear of violent death was the mundane psychological backdrop to everyday existence. One of the most effective chapters here explores the phenomenon of plague through London's bills of mortality. Indeed, the bad weather of the 1590s, which kept the plague at bay and the theatres booming, may have given Shakespeare just the opportunity he needed to get started.

But, pace MacGregor, this is not "a three-way conversation between the objects themselves, the people who used or looked at them and the words of the playwright" At best, the interrogation of these objects yields a sequence of fascinating footnotes to Shakespeare's timeless poetry. If you want a "three-way conversation" between Shakespeare, his work and the present day, read Stephen Greenblatt, a Shakespeare scholar unaccountably not cited here.

Almost as a kind of afterthought, one of MacGregor's concluding "objects" is a copy of the First Folio. Here, finally, he asks the question that lets light into the eternal conundrum of these plays: "How has this supremely public writer become the private companion of so many, his words the stuff that their hopes, fears and dreams are made on?"

The answer, to which MacGregor's inventory is a picturesque complement, is through the subtle interplay of character, language and plot. As Simon Russell Beale put it in his 2009 Ernest Jones lecture, acting Shakespeare is "three-dimensional literary criticism". But that's about words and ideas, not things.